proving $\mathbb{Q}^2$ is regular space in separately open topology.

Question

I am studying the proof of theorem 5.5, regularity of the topology of separate continuity

We are familiar with Euclidean topology on $\mathbb{R}^2$, in which open sets are defined with respect to the $\epsilon$ balls centred at a point in the plane. Analogous to these $\epsilon$ balls, we define the '$\epsilon-$ plus' centred at a point $(a, b)$ as $$ P_\epsilon (a, b) = \{ (x, b) \in \mathbb{R}^2 : |x - a| < \epsilon \} \cup \{ (a,y) \in \mathbb{R}^2 : |y - b | < \epsilon \} .$$ We say $ U \subset \mathbb{R}^2$ is separately open set, if for each point in $U$, there is some $\epsilon > 0 $ such that $P_\epsilon (a,b) \subset U$. These separately open sets form a topology on $\mathbb{R}^2$, called separately open topology. It is denoted as $\mathbb{R \otimes R}$. when we restrict it to $\mathbb{Q}^2$ we get the topology $\mathbb{Q \otimes Q}.$ In particular, I want to show that $\mathbb{Q \otimes Q}$ is regular.

Now, in the proof of theorem 5.5; i do not understand the notation and what they mean by $(\gamma)$ and $(\delta)$.

In the construction given in second paragraph, I have following doubts:

1. $E_\epsilon$ cover the closed set A. and since A is closed we can find the countable number of closed $A_j \subset A$, such that $A = \cup_j A_j$. but in the construction above A is a Clopen set. please explain it.

2. how is the $diam(R_j) < 1/j$ can satisfy $(R_j \times A_j) \cap K_{n-1} = \phi$?

Answer 1

The space $X\otimes Y$ and the notations $E_x,E^y$ were defined in Definition 1.1 of this paper. $S^∘$ denotes the interior of $S.$ This answers your first question "i do not understand the notation and what they mean by $(γ)$ and $(δ)$".

As for your (twofold) second question:

1. The partition of $Y$ into two clopen subsets $A,B$ such that $(H_{n-1})_x\subseteq A$ and $(K_{n-1})_x\subseteq B$ is prior to the definition of the $E_\varepsilon$'s. The existence of such a partition is due to the fact that $(H_{n-1})_x,(K_{n-1})_x$ are closed and disjoint (since $H_{n-1},K_{n-1}$ are, by induction hypothesis), and $Y$ is strongly $0$-dimensional.
2. Since $\inf\{d(x,(K_{n-1})^y)\mid y\in A_j\}>0,$ for every $x'$ sufficiently close to $x$ and every $y\in A_j$, we have $(x',y)\notin K_{n-1}.$ Now, $x$ has a basis of clopen neighborhoods, so this "$x'$ sufficiently close to $x$" can be replaced by $x'\in R_j$, for some "small" (e.g. with diameter $<1/j$) clopen neighborhood $R_j$ of $x$.